\subsection{PD Control Experiment}
This section investigates the effect of different values of $K_p$ and $K_d$ on the motion of the base joint ($q_1$). The base joint will be moved through a step of 0.5 radians while fixing the two planar joints. 

As Kp increases from 1 to 5, the output exhibits more oscillation and overshoot as shown in Figures \ref{fig:pd_1_0} to \ref{fig:pd_5_0}. The motor torque also increases as the proportional gain increases. In Figures \ref{fig:pd_5_01} to \ref{fig:pd_5_05}, it is observed that adding the derivative component (Kd) helps to reduce the overshoot and oscillations.  However, there are still steady state errors. In Figure \ref{fig:pd_9_05}, the steady state error is reduced by having a proportional gain of 9. The observation is that a higher proportional gain improves steady state tracking.

\begin{figure}[htpb]
  \begin{center}
    \includegraphics[width=10cm]{1_0}
  \end{center}
  \caption{Kp = 1, Kd = 0}
  \label{fig:pd_1_0}
\end{figure}
\begin{figure}[htpb]
  \begin{center}
    \includegraphics[width=10cm]{2_0}
  \end{center}
  \caption{Kp = 2, Kd = 0}
  \label{fig:pd_2_0}
\end{figure}
\begin{figure}[htpb]
  \begin{center}
    \includegraphics[width=10cm]{5_0}
  \end{center}
  \caption{Kp = 5, Kd = 0}
  \label{fig:pd_5_0}
\end{figure}
\begin{figure}[htpb]
  \begin{center}
    \includegraphics[width=10cm]{5_01}
  \end{center}
  \caption{Kp = 5, Kd = 0.1}
  \label{fig:pd_5_01}
\end{figure}
\begin{figure}[htpb]
  \begin{center}
    \includegraphics[width=10cm]{5_05}
  \end{center}
  \caption{Kp = 5, Kd = 0.5}
  \label{fig:pd_5_05}
\end{figure}
\begin{figure}[htpb]
  \begin{center}
    \includegraphics[width=10cm]{9_05}
  \end{center}
  \caption{Kp = 9, Kd = 0.5}
  \label{fig:pd_9_05}
\end{figure}

